Galaxies are complex systems the evolution of which apparently results from the interplay of dynamics, star formation, chemical enrichment, and feedback from supernova explosions and supermassive black holes. The hierarchical theory of galaxy formation holds that galaxies are assembled from smaller pieces, through numerous mergers of cold dark matter. The properties of an individual galaxy should be controlled by six independent parameters including mass, angular-momentum, baryon-fraction, age and size, as well as by the accidents of its recent haphazard merger history. Here we report that a sample of galaxies that were first detected through their neutral hydrogen radio-frequency emission, and are thus free of optical selection effects, shows five independent correlations among six independent observables, despite having a wide range of properties. This implies that the structure of these galaxies must be controlled by a single parameter, although we cannot identify this parameter from our dataset. Such a degree of organisation appears to be at odds with hierarchical galaxy formation, a central tenet of the cold dark matter paradigm in cosmology.